Theorem upfval 49084

Description: Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) (Proof shortened by Zhi Wang, 12-Nov-2025.)

Hypotheses

Ref	Expression
upfval.b	⊢ 𝐵 = (Base‘𝐷)
upfval.c	⊢ 𝐶 = (Base‘𝐸)
upfval.h	⊢ 𝐻 = (Hom ‘𝐷)
upfval.j	⊢ 𝐽 = (Hom ‘𝐸)
upfval.o	⊢ 𝑂 = (comp‘𝐸)

Assertion

Ref	Expression
upfval	⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Distinct variable groups:   𝐵,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐶,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐷,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐸,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐻,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐽,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝑂,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦

Proof of Theorem upfval

Dummy variables 𝑏 𝑐 𝑑 𝑒 ℎ 𝑗 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6880	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) ∈ V)
2		fveq2 6865	. . . . . 6 ⊢ (𝑑 = 𝐷 → (Base‘𝑑) = (Base‘𝐷))
3	2	adantr 480	. . . . 5 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = (Base‘𝐷))
4		upfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
5	3, 4	eqtr4di 2783	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = 𝐵)
6		fvexd 6880	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) ∈ V)
7		simplr 768	. . . . . . 7 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑒 = 𝐸)
8	7	fveq2d 6869	. . . . . 6 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = (Base‘𝐸))
9		upfval.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
10	8, 9	eqtr4di 2783	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = 𝐶)
11		fvexd 6880	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) ∈ V)
12		simplll 774	. . . . . . . 8 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → 𝑑 = 𝐷)
13	12	fveq2d 6869	. . . . . . 7 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = (Hom ‘𝐷))
14		upfval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
15	13, 14	eqtr4di 2783	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = 𝐻)
16		fvexd 6880	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) ∈ V)
17		simp-4r 783	. . . . . . . . 9 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → 𝑒 = 𝐸)
18	17	fveq2d 6869	. . . . . . . 8 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = (Hom ‘𝐸))
19		upfval.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐸)
20	18, 19	eqtr4di 2783	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = 𝐽)
21		fvexd 6880	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) ∈ V)
22		simp-5r 785	. . . . . . . . . 10 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → 𝑒 = 𝐸)
23	22	fveq2d 6869	. . . . . . . . 9 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = (comp‘𝐸))
24		upfval.o	. . . . . . . . 9 ⊢ 𝑂 = (comp‘𝐸)
25	23, 24	eqtr4di 2783	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = 𝑂)
26		simp-6l 786	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑑 = 𝐷)
27		simp-6r 787	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑒 = 𝐸)
28	26, 27	oveq12d 7412	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑑 Func 𝑒) = (𝐷 Func 𝐸))
29		simp-4r 783	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑐 = 𝐶)
30		simp-5r 785	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑏 = 𝐵)
31	30	eleq2d 2815	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ 𝐵))
32		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑗 = 𝐽)
33	32	oveqd 7411	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑥)) = (𝑤𝐽((1st ‘𝑓)‘𝑥)))
34	33	eleq2d 2815	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥)) ↔ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))))
35	31, 34	anbi12d 632	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥)))))
36	32	oveqd 7411	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑦)) = (𝑤𝐽((1st ‘𝑓)‘𝑦)))
37		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ℎ = 𝐻)
38	37	oveqdr 7422	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
39		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑜 = 𝑂)
40	39	oveqd 7411	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦)) = (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦)))
41	40	oveqd 7411	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))
42	41	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
43	38, 42	reueqbidv 3400	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
44	36, 43	raleqbidv 3322	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
45	30, 44	raleqbidv 3322	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
46	35, 45	anbi12d 632	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))))
47	46	opabbidv 5181	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})
48	28, 29, 47	mpoeq123dv 7471	. . . . . . . 8 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
49	21, 25, 48	csbied2 3907	. . . . . . 7 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
50	16, 20, 49	csbied2 3907	. . . . . 6 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → ⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
51	11, 15, 50	csbied2 3907	. . . . 5 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
52	6, 10, 51	csbied2 3907	. . . 4 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
53	1, 5, 52	csbied2 3907	. . 3 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
54		df-up 49082	. . 3 ⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))
55		ovex 7427	. . . 4 ⊢ (𝐷 Func 𝐸) ∈ V
56	9	fvexi 6879	. . . 4 ⊢ 𝐶 ∈ V
57	55, 56	mpoex 8067	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) ∈ V
58	53, 54, 57	ovmpoa 7551	. 2 ⊢ ((𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
59		reldmup 49083	. . . 4 ⊢ Rel dom UP
60	59	ovprc 7432	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = ∅)
61		reldmfunc 48992	. . . . . 6 ⊢ Rel dom Func
62	61	ovprc 7432	. . . . 5 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 Func 𝐸) = ∅)
63	62	orcd 873	. . . 4 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → ((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅))
64		0mpo0 7479	. . . 4 ⊢ (((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
65	63, 64	syl 17	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
66	60, 65	eqtr4d 2768	. 2 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
67	58, 66	pm2.61i 182	1 ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3046  ∃!wreu 3355  Vcvv 3455  ⦋csb 3870  ∅c0 4304  ⟨cop 4603  {copab 5177  ‘cfv 6519  (class class class)co 7394   ∈ cmpo 7396  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185  Hom chom 17237  compcco 17238   Func cfunc 17822   UP cup 49081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-func 17826  df-up 49082

This theorem is referenced by:  upfval2  49085  reldmup2  49089  relup  49090  uprcl  49091